Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $17.4$ years; the standard deviation is $3.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $10.4$ and $27.9$ years.
Explanation: $17.4$ $13.9$ $20.9$ $10.4$ $24.4$ $6.9$ $27.9$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $17.4$ years. We know the standard deviation is $3.5$ years, so one standard deviation below the mean is $13.9$ years and one standard deviation above the mean is $20.9$ years. Two standard deviations below the mean is $10.4$ years and two standard deviations above the mean is $24.4$ years. Three standard deviations below the mean is $6.9$ years and three standard deviations above the mean is $27.9$ years. We are interested in the probability of a seal living between $10.4$ and $27.9$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of seals between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular seal living between $10.4$ and $27.9$ years is ${95\%} + \color{orange}{2.35\%}$, or $97.35\%$.